The three-digit integer $63\underline{\hphantom{0}}$ is a multiple of 3. What is the greatest possible difference between two of the possibilities for the units digit?
The integer is a multiple of three if the sum of its digits is a multiple of three. Since 6 and 3 are both multiples of three, the units digit must also be a multiple of three. The possibilities for this digit are 0, 3, 6, or 9. The greatest possible difference between any two possibilities is $9-0=\boxed{9}$.